Learning the Clustering of Longitudinal Shape Data Sets into a Mixture of Independent or Branching Trajectories

4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of objects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered. Also, different deforming objects, hereinafter referred to as 4D surfaces, evolve at different speeds and thus they need to be temporally aligned. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemannian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields (SRNF) where the [Formula: see text] metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces even in the presence of large elastic deformations and large variations in their execution rates; (2) the computation of geodesics between 4D surfaces; (3) the computation of statistical summaries, such as means and modes of variation, of collections of 4D surfaces; and (4) the synthesis of random 4D surfaces. We demonstrate the performance of the proposed framework using 4D facial surfaces and 4D human body shapes.

Feasibility of a longitudinal statistical atlas model to study aortic growth in congenital heart disease

Studying anatomical shape progression over time is of utmost importance to refine our understanding of clinically relevant processes. These include vascular remodeling, such as aortic dilation, which is particularly important in some congenital heart defects (CHD). A novel methodological framework for three-dimensional shape analysis has been applied for the first time in a CHD scenario, i.e., bicuspid aortic valve (BAV) disease, the most common CHD. Three-dimensional aortic shapes (n = 94) reconstructed from cardiovascular magnetic resonance imaging (MRI) data as surface meshes represented the input for a longitudinal atlas model, using multiple scans over time (n = 2-4 per patient). This model relies on diffeomorphism transformations in the absence of point-to-point correspondence, and on the right combination of initialization, estimation and registration parameters. We computed the shape trajectory of an average disease progression in our cohort, as well as time-dependent parameters, geometric variations and the average shape of the population. Results cover a spatiotemporal spectrum of visual and numerical information that can be further used to run clinical associations. This proof-of-concept study demonstrates the feasibility of applying advanced statistical shape models to track disease progression and stratify patients with CHD.

New confinement index and new perspective for comparing countries - COVID-19

BACKGROUND AND OBJECTIVE

In the difficult problem of comparing countries regarding their lockdown measures or deaths caused by the COVID-19, there is still no agreement on what is the best strategy to follow. Thus, we propose a new way of comparison countries that avoids the main difficulties in the comparison by using three-dimensional trajectories for this type of data.

METHODS

We introduce a new index to analyze the level of confinement that each country was subject to overtime, based on the Community Mobility Reports published by Google resorting to Principal Component Analysis. Subsequently, by using longitudinal clustering, we divide the European countries into similar groups according to the COVID-19 obits and also to the confinement index. However, to make the most out of the clustering methods we resort to artificial longitudinal data to evaluate both the methods and the indices.

RESULTS

By using artificial data, we discover that Calinski-Harabasz outperformed other internal indices in indicating the real number of clusters. The tests also suggested that K-means with Euclidean distance was the best method among the ones studied. With the application to both the mobility and fatalities datasets, we found two groups in each one.

CONCLUSIONS

Our analysis enables us to discover that European northern countries had more mobility during the first confinement and that the deaths caused by COVID-19 started to drop around the 40th day since the first death.

Understanding the Variability in Graph Data Sets through Statistical Modeling on the Stiefel Manifold

Network analysis provides a rich framework to model complex phenomena, such as human brain connectivity. It has proven efficient to understand their natural properties and design predictive models. In this paper, we study the variability within groups of networks, i.e., the structure of connection similarities and differences across a set of networks. We propose a statistical framework to model these variations based on manifold-valued latent factors. Each network adjacency matrix is decomposed as a weighted sum of matrix patterns with rank one. Each pattern is described as a random perturbation of a dictionary element. As a hierarchical statistical model, it enables the analysis of heterogeneous populations of adjacency matrices using mixtures. Our framework can also be used to infer the weight of missing edges. We estimate the parameters of the model using an Expectation-Maximization-based algorithm. Experimenting on synthetic data, we show that the algorithm is able to accurately estimate the latent structure in both low and high dimensions. We apply our model on a large data set of functional brain connectivity matrices from the UK Biobank. Our results suggest that the proposed model accurately describes the complex variability in the data set with a small number of degrees of freedom.

On the curved exponential family in the Stochastic Approximation Expectation Maximization Algorithm

The Expectation-Maximization Algorithm (EM) is a widely used method allowing to estimate the maximum likelihood of models involving latent variables. When the Expectation step cannot be computed easily, one can use stochastic versions of the EM such as the Stochastic Approximation EM. This algorithm, however, has the drawback to require the joint likelihood to belong to the curved exponential family. To overcome this problem, [16] introduced a rewriting of the model which “exponentializes” it by considering the parameter as an additional latent variable following a Normal distribution centered on the newly defined parameters and with fixed variance. The likelihood of this new exponentialized model now belongs to the curved exponential family. Although often used, there is no guarantee that the estimated mean is close to the maximum likelihood estimate of the initial model. In this paper, we quantify the error done in this estimation while considering the exponentialized model instead of the initial one. By verifying those results on an example, we see that a trade-off must be made between the speed of convergence and the tolerated error. Finally, we propose a new algorithm allowing a better estimation of the parameter in a reasonable computation time to reduce the bias.